Can you prove $(x-y)^3+(y-z)^3+(z-x)^3 = 3(x-y)(y-z)(z-x)$? Show that $$(x-y)^3+(y-z)^3+(z-x)^3 = 3(x-y)(y-z)(z-x)$$
This can be shown through expansion but there is a more elegant solution
I cannot discover anything I would consider elegant. Can anyone help?
 A: $$ a^3 + b^3 + c^3 - 3 abc = (a+b+c)(a^2 + b^2 + c^2 - bc - ca - ab) $$
Note that this is the determinant of
$$ M \; = \;  
 \left(  \begin{array}{rrr}
  a  &  b  &  c \\
  c   &  a  &  b \\
  b  &  c   &  a   
\end{array} 
  \right),  
  $$
the matrix being evidently singular when $a=b=c$ or when $a+b+c=0.$
Meanwhile, with the evident possibility of permuting the letters, 
$$  (a^2 + b^2 + c^2 - bc - ca - ab)  = \frac{1}{4} \left( \, (a+b-2c)^2 + 3 (a-b)^2 \, \right)    $$
 and so is positive semidefinite only, it comes out to $0$ when $a=b=c.$
A: It's simply  $\ a^3\!+b^3\!-(a\!+\!b)^3=\,-3ab(a\!+\!b)\ $ for $\ a,b\, =\, x\!-\!y,\ y\!-\!z$.
A: If one believes that the polynomial ring in 3 variables is a unique factorization ring, which it is, then we relatively easily see that both sides vanish when $x=y$ or when $x=z$ or when $y=z$. The constant $3$ can be determined by plugging in $x=0$, $y=1$, and $z=-1$, for example.
A: $$a^3+b^3+c^3-3abc=(a+b)^3-3ab(a+b)+c^3-3abc$$
$$=(a+b)^3+c^3-3ab(a+b)-3abc$$
$$=[(a+b)+c][(a+b)^2-(a+b)c+c^2]-3ab[(a+b)+c]$$
$$=(a+b+c)[(a+b)^2-(a+b)c+c^2-3ab]$$
$$=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)$$
Though not required in the current case, $$a^2+b^2+c^2-ab-bc-ca=\frac{(a-b)^2+(b-c)^2+(c-a)^2}2$$ which will be zero iff $\displaystyle a-b=b-c=c-a=0\implies ?$
